Implementation and Analysis of Various Kalman Filtering Techniques for Target Tracking

Since measurements of process variables are subject to measurements errors as well as process variability, data reconciliation is the procedure of optimally adjusting measured date so that the adjusted values obey the conservation laws and constraints. Thus, data reconciliation for dynamic systems is fundamental and important for control, fault detection, and system optimization. Attempts to successfully implement estimators are often hindered by serve process nonlinearities, complicated state constraints, and un-measurable perturbations. As a constrained minimization problem, the dynamic data reconciliation is dynamically carried out to product smoothed estimates with variances from the original data. Many algorithms are proposed to solve such state estimation such as the extended Kalman filter (EKF), the unscented Kalman filter, and the cubature Kalman filter (CKF). In this paper, we investigate the use of CKF algorithm in comparative with the EKF to solve the nonlinear dynamic data reconciliation problem. First we give a broad overview of the recursive nonlinear data dynamic reconciliation (RNDDR) scheme, then present an extension to the CKF algorithm, and finally address the issue of how to solve the constraints in the CKF approach. The CCRNDDR method is proposed by applying the RNDDR in the CKF algorithm to handle nonlinearity and algebraic constraints and bounds. As the sampling idea is incorporated into the RNDDR framework, more accurate estimates can obtained via the recursive nature of the estimation procedure. The performance of the CKF approach is compared with EKF and RNDDR on nonlinear process systems with constraints. The conclusion is that with an error optimization solution of the correction step, the reformulated CKF shows high performance on the selection of nonlinear constrained process systems. Simulation results show the CCRNDDR is an efficient, accurate and stable method for real-time state estimation for nonlinear dynamic processes.

Performance evaluation of Cubature Kalman filter in a GPS/IMU tightly-coupled navigation system

Data fusion algorithms are inevitable in improving the accuracy and reliability of radar target tracking. Considering the shortcomings of target tracking in Cartesian coordinates, all the fusion process are carried out in spherical coordinate system. To fuse data from bidimensional radar and Infrared Search and Track (IRST), five data fusion methods, Bayesian fusion, Signal variance based fusion, Optimal fusion, Error variance based fusion and Extended Kalman Filter (EKF) based fusion are proposed and analyzed. The performances of the methods are analyzed in terms of Root Mean Square Error in Position (RMSPE) and Mean Absolute Error in Position (MAPE) for a wide range of measurement noise variances. It is concluded that EKF based fusion has the best error performance. Kalman Filter State-Vector Fusion (KF-SVF) which is a hierarchical fusion architecture, having first level of fusion with Kalman filter and second level with error variance based fusion method, is available in the literature. In this work it has been expanded to Extended Kalman Filter—State-Vector Fusion (EKF-SVF) with either error variance based fusion, signal variance based fusion or optimal information fusion as second level. All the three EKF-SVF methods have similar error characteristics, but EKF-SVF with signal variance based fusion, unlike the other two, provides less computational cost since it does not involve matrix inversion. While comparing EKF based fusion and EKF-SVF, EKF-SVF offers increased error minimization.

In this paper, vision-based relative navigation of two spacecraft is addressed using the sparse Gauss-Hermite quadrature filter. The relative navigation provides the estimation of the relative attitude and relative orbit between spacecraft, which is a challenging filtering problem since it is a highly nonlinear and high dimensional estimation problem. Many filters have been used to solve this problem, such as, the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). However, these filters are not accurate enough when the initial uncertainty is large or the nonlinearity is significant. Moreover, although the conventional Gauss-Hermite quadrature filter is more accurate than the UKF, it cannot be used since a huge number of points is required. It is shown in this paper that the sparse Gauss-Hermite quadrature filter can achieve better estimation accuracy than the EKF, the UKF, and the cubature Kalman filter without excessive computation load.

A novel sparse Gauss―Hermite quadrature filter is proposed using a sparse-grid method for multidimensional numerical integration in the Bayesian estimation framework. The conventional Gauss―Hermite quadrature filter is computationally expensive for multidimensional problems, because the number of Gauss―Hermite quadrature points increases exponentially with the dimension. The number of sparse-grid points of the computationally efficient sparse Gauss―Hermite quadrature filter, however, increases only polynomially with the dimension. In addition, it is proven in this paper that the unscented Kalman filter using the suggested optimal parameter is a subset of the sparse Gauss―Hermite quadrature filter. The sparse Gauss-Hermite quadrature filter is therefore more flexible to use than the unscented Kalman filter in terms of the number of points and accuracy level, and it is more efficient than the conventional Gauss―Hermite quadrature filter. The application to the spacecraft attitude estimation problem demonstrates better performance of the sparse Gauss―Hermite quadrature filter in comparison with the extended Kalman filter, the cubature Kalman filter, and the unscented Kalman filter.

In this paper, the state trajectories of a hypersonic vehicle is estimated using a new generalized cubature Kalman filter. This new nonlinear estimation technique is based on a class of arbitrary-degree cubature rules to compute the multivariate integrals in the Bayesian filtering algorithm and can achieve higher degree of estimation accuracy. The performance of this new cubature Kalman filter is compared with the Extended Kalman filter, the unscented Kalman filter and the conventional third-degree cubature Kalman filter. The simulation results demonstrate that the proposed filter outperforms those three filters in estimation accuracy and convergence rate. It is also computationally more efficient than many other point-based estimation techniques.

The trajectory estimation of a highly maneuvering target is a challenging problem and has practical applications. The interacting multiple model (IMM) filter is a well-established filtering algorithm for the trajectory estimation of maneuvering targets. In this study, we present an overview of IMM filtering algorithms for tracking a highly-maneuverable fighter aircraft using an air moving target indicator (AMTI) radar on another aircraft. This problem is a nonlinear filtering problem due to nonlinearities in the dynamic and measurement models. We first describe single-model nonlinear filtering algorithms: the extended Kalman filter (EKF), unscented Kalman filter (UKF), and cubature Kalman filter (CKF). Then, we summarize the IMM-based EKF (IMM-EKF), IMM-based UKF (IMM-UKF), and IMM-based CKF (CKF). In order to compare the state estimation accuracies of the IMM-based filters, we present a derivation of the posterior Cramér-Rao lower bound (PCRLB). We consider fighter aircraft traveling with accelerations 3g, 4g, 5g, and 6g and present numerical results for state estimation accuracy and computational cost under various operating conditions. Our results show that under normal operating conditions, the three IMM-based filters have nearly the same accuracy. This is due to the accuracy of the measurements of the AMTI radar and the high data rate.

According to the problem on tracking planning of mobility assistance robot, a tracking planning method was presented using the kalman filter (KF). Based on known nonholonomic dynamic model of mobility assistance robot, the cubature kalman filter (CKF) was used to improve the influence of nonlinear system, and the autonomous navigation and control of mobility assistance robot was realized. The experimental results indicate that the method presented in this paper was excellent than extended kalman filter (EKF) and unscented kalman filter (UKF) in the stability, precision and computational complexity.

The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.

Sparse-grid quadrature nonlinear filtering

An introduction of a new observer for estimating speed and flux of induction motor drive is done in this paper. The main tool of designing the observer is cubature Kalman filter (CKF). In order to compute multivariate integrals for moment estimation the CKF uses spherical-radial transformation followed by a third degree spherical-radial rule. The utmost advantage of CKF lies on capturing the system nonlinearity using the spherical-radial transformation giving it and edge over Extended Kalman Filter (EKF), as the errors is in it are minimized by linearizing the state model of the system. In this paper square root form of CKF (SRCKF) is used to eliminate the most hazardous situation of the loss of positive definiteness of error covariance matrix in each update which can stop the CKF from running continuously. The sensor less direct torque control is used for an Induction motor of 0.75Kw, 1475 rpm, for conducting performance test of the proposed observer. For simpler design of SRCKF, state space model of IM in stationary reference frame has been used. Effectiveness of the proposed observer is validated using simulation result for a VSI fed IM drive and compared with that of an EKF observer.

This paper presents a Cubature Kalman Filter (CKF) based speed and position observer for a Permanent Magnet Synchronous Motor (PMSM). CKF is a new variety of Kalman filter which uses a third degree spherical-radial cubature rule to numerically compute multivariate moment integrals. Unlike in an Extended Kalman Filter(EKF), mean and covariance are propagated through the non-linear system, which minimizes the errors due to linearization. The proposed observer is incorporated in the sensorless control of an IPMSM of 1.5kW, 3000rpm rating. For the CKF algorithm, PMSM is modeled in stationary αβ reference frame. To get comparatively better transient performance and convergence of the CKF for a non-zero initial rotor position, system covariance matrix Q is chosen adaptively. Simulation results for a VSI fed IPMSM are presented and the convergence of CKF is shown for a variation of stator resistance. Performance of the proposed observer is compared with that of an EKF observer.

Multiple Target Tracking (MTT) is one of the most challenging topics in radar target tracking. In addition to conventional position measurements, the integration of target Doppler and power information can offer valuable insights into a target’s kinematic state, thereby improving tracking performance. This article discusses a tracking approach that incorporates those components into the measurement enhancement process and assesses the advantages of the proposed strategy. Firstly, we investigated an improved data association scheme that exploits statistical features of position, Doppler and power. The estimated Doppler value calculated from the target range and timestamp will be compared with the measured Doppler to deal with the Doppler ambiguity situation. Secondly, we studied an augmented unscented Kalman filter (UKF) algorithm using position, Doppler measures, and the indirect measure of radial velocity in the linear domain. The experimental results show that the proposed solution has good performance in terms of reduced number of false tracks and improving the accuracy of the target state estimation.

In this paper, according to the characteristics of underwater target tracking, extended Kalman filter (EKF) algorithm was applied to address underwater bearing-only non-maneuverable target tracking problem. To ensure the observability in bearing-only passive target tracking, we use two sensor arrays in uniform rectilinear motion to track the single target. EKF is recursive Bayesian filter algorithm based on the linearization of the nonlinearities in the state and the measurement system. And the simulation experiment shows that, to the constant velocity single target, the bearings-only EKF algorithm with dual-array has good results, also the EKF has lower computation complexity than unscented Kalman filter and particle filter. It confirms the effectiveness of the EKF algorithm in solving the underwater bearing-only target tracking problem.

Simultaneous state and parameter estimation is essential for control system design and dynamic modeling of physical systems. This capability provides critical real-time insight into system behavior, supports the discovery of underlying mechanisms, and facilitates adaptive control strategies. Surveyed in this review paper are two classes of state and parameter estimation methods: Kalman Filters and Luenberger Observers. The Kalman Filter framework, including its major variants such as the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Cubature Kalman Filter (CKF), and Ensemble Kalman Filter (EnKF), has been widely applied for joint and dual estimation in linear and nonlinear systems under uncertainty. In parallel, Luenberger observers, typically used in deterministic settings, offer alternative approaches through high-gain, sliding mode, and adaptive observer structures. This review focuses on the theoretical foundations, algorithmic developments, and application domains of these methods and provides a comparative analysis of their advantages, limitations, and practical relevance across diverse engineering scenarios.